The book of Hebrews in the New Testament shares a name with the Hebrew people of the Mosaic Law. Why was that book named Hebrews?
It's called that because it was written as a letter to Hebrews. Most of the Epistles are titled after the group they were written to. (Corinthians was written to the Church at Corinth, etc.)
Hebrews was written before the fall of Jerusalem and the destruction of the Temple in 70 A.D. Written To:
Hebrew Christians who were wavering in their faith and all future readers of the Bible.
There's more similar information all over, including Wikipedia.
It seems that one reason that this book was named Hebrews was due to:
the earliest form of the text that has come down to us, P46, this book had the title (Greek won't display here) - (Pros Hebraious, "To[the]Hebrews").
Carson, D. A., and Douglas J. Moo. "Hebrews." An Introduction to the New Testament. Grand Rapids, MI: Zondervan, 2005. 609. Print.
Although authorial attribution is missing there seems to be a high likelihood of this being a possible original title. I am no scholar though so I am just throwing this in as food for thought.
This letter was written to Hebrews (Judean believers of Jesus Christ from Judea). Hebrews is the original name of Judeans (or later called "Jews" in English) due to the fact that they are the descendants of Heber.
Josephus' Antiquities of Jews Book 1, Chapter 6, Paragraph 4 - "Sala was the son of Arphaxad; and his son was Heber, from whom they originally called the Jews Hebrews."
Hebrews became known as Judeans or Jews after they came back from Babylon.
Josephus' Antiquities of Jews Book 11:5:7 - "So the Jews prepared for the work: That is the name they are called by from the day that they came up from Babylon, which is taken from the tribe of Judah, which came first to these places, and thence both they and the country gained that appellation."
Apostle Paul calls himself a Hebrew.
Philippians 3:5 (NIV) - "circumcised on the eighth day, of the people of Israel, of the tribe of Benjamin, a Hebrew of Hebrews; in regard to the law, a Pharisee."
Jewish Priest Josephus also calls himself "a Hebrew."
Jewish Wars (Book 1, Preface, Paragraph 1)- "I have proposed to myself, for the sake of such as live under the government of the Romans, to translate those books into the Greek tongue, which I formerly composed in the language of our country, and sent to the Upper Barbarians. Joseph, the son of Matthias, by birth a Hebrew, a priest also, and one who at first fought against the Romans myself, and was forced to be present at what was done afterwards, [am the author of this work]."
(I made a substantial change to this answer since first posting it. I apologize for my initial confusion.]
It is called the Book of Hebrews because the one who titled it recognized something I believe the Holy Spirit is trying to reveal to all the followers of Messiah Jesus today. Namely, that all of us - Jew and "gentile" alike - must be Hebrews! That is, in our human lineage we must all be physical seed of Abraham! Otherwise the letter would not apply to us and would be speaking only to the Jewish followers of Jesus. Gentile believers would be excluded. And that is an impossibility since every book in the NT is speaking to the entire body of Messiah. There are no promises just for the Jews. Or just for the so-called gentile believers. "For all the promises have their amen in Jesus" (2 Cor. 1:20).
This revelation of our true lineage, although affirmed throughout the letter, is seen most clearly in Hebrews 2:11-16. We are informed that "both he who sanctifies (Jesus) and those who are sanctified (all who are his, both Jew and non-Jew) are "all from one Father" (italicized because not in text); for which reason he is not ashamed to call them brethren." The "one" who is the father of all believers is revealed in the OT quotes that follow to be our Heavenly Father - God Himself. As "children of God" we are all spiritual brethren. But the text goes even further and gives another reason for Jesus to call us brethren. For we are told he "partook of the same flesh and blood." That "he had to be made like his brethren in all things." In other words we all have the same earthly father as well. And he is identified in verse 16 as Abraham. "He [Jesus] only gives help to the seed of Abraham." This eliminates Adam as the father who gives us our commonality of the flesh that makes us all brothers.
If this seems a bit shocking it's only because we forget that the hope of Abraham was to become "the [physical] father of many gentiles/nations" (Rom. 4:18). That's why God changed his name from Abram (meaning "exalted father) to Abraham (meaning "father of many gentiles/nations"). We know his seed was scattered all over the world when ten of the twelve tribes were exiled by Assyria in 722 BC after God turned them into "gentiles" for their idolatry (see Hos.1. 2 Kings 17:18). So the ones whom the Father leads to Jesus for salvation (John 6:37,44), who are not Jewish, must be that scattered seed. Once this understanding is gained it is readily seen throughout Scripture, unifying the whole Bible into one story about the promises made to one man - Abraham. Galatians 3:29 says it most succinctly: "If you belong to Messiah, you ARE Abraham's seed, heirs according to promise."